Download Conjugation coinvariants of quantum matrices

Conjugation coinvariants of quantum matrices
M. Domokos ∗ and T. H. Lenagan
Abstract
A quantum deformation of the classical conjugation action of GL(N, C) on the
space of N × N matrices M (N, C) is defined via a coaction of the quantum general linear group O(GLq (N, C)) on the algebra of quantum matrices O(Mq (N, C)).
The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of O(Mq (N, C)) generated by (weighted) sums of principal quantum
minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.
2000 Mathematics Subject Classification. 16P40, 16W30, 16W35, 16S15, 13A50, 20G42.
Keywords: coinvariant, quantum general linear group, quantum matrices, quantum determinant, principal minors.
1
Introduction
The conjugation A → g −1 Ag of a complex N × N matrix A by an invertible matrix g
induces an action of GL(N, C) on the coordinate algebra C[X ij ] of N × N matrices. The
invariant functions with respect to this action are well-known, they are the elements in the
C-subalgebra of C[Xij ] that is generated by the trace functions σi , for i = 1, . . . , N, where
σi is the sum of the i × i principal minors.
We start this paper by developing a quantum analogue of this conjugation action. The
natural way to proceed is to operate at the level of coordinate algebras. At this level
the conjugation action of GL(N, C) translates into a coaction of the coordinate algebra
of the general linear group, O(GL(N, C)), on the coordinate algebra of N × N matrices,
O(M (N, C)) and the problem of finding invariants becomes the problem of finding the
coinvariants of this coaction, where a is a coinvariant if a 7→ a ⊗ 1. In order to quantize
the action, we need to operate with the algebra of quantum matrices, O(M q (N, C)) and
the quantum general linear group, O(GLq (N, C)). We start by recalling the definitions of
these algebras and fixing our notation, since notation varies within the literature.
The algebra of N × N quantum matrices over C is the C-algebra generated by N 2
indeterminates xij , for i = 1, . . . , N , subject to the following relations.
∗
Partially supported by OTKA No. F 32325 and T 34530. Most of this research was done during a
visit to Edinburgh with the support of the London Mathematical Society and during the first author’s
European Community Marie Curie Fellowship held in Edinburgh.
1
xij xil
xij xkj
xil xkj
xij xkl − xkl xij
=
=
=
=
qxil xij ,
qxkj xij ,
xkj xil ,
(q − q −1 )xil xkj ,
for 1 ≤ i < k ≤ N and 1 ≤ j < l ≤ N , where q ∈ C∗ .
The algebra O(Mq (N, C)) can be presented as an iterated Ore extension, and so it is
easily seen to be a noetherian domain.
The quantum determinant, det q , is defined by
X
(−q)l(σ) x1,σ(1) . . . xN,σ(N ) .
detq :=
σ∈SN
From [7, Theorem 4.6.1], we know that det q is a central element in O(Mq (N, C)), and so it
may be inverted. The resulting algebra is the quantum general linear group, O(GL q (N, C)),
so that
O(GLq (N, C)) := O(Mq (N, C))[detq−1 ].
The algebra O(GLq (N, C)) is a Hopf algebra, and we denote the antipode of this algebra
by S. We will often denote the element xij by uij when we are considering it as an element
of O(GLq (N, C)) rather than as an element of O(Mq (N, C)).
The reader should be aware that in many papers the rôles of q and q −1 are interchanged,
and so one has to be careful in translating results from one paper to another.
2
Co-commutative elements as coinvariants
Let A be a Hopf algebra with multiplication µ, comultiplication ∆, counit ε, and antipode
S. We define a right coaction α of A on itself by
X
α : A → A ⊗ A, h 7→
h2 ⊗ h3 S(h1 ),
(1)
where we are using Sweedler’s notation in the defining formula.
In the present paper we call α the adjoint coaction, see Section 7 for further comments
on this definition and terminology.
Recall that an element h of A is said to be co-commutative if τ ∆(h) = ∆(h), where τ
denotes the flip.
Theorem 2.1 The element h is co-commutative if and only if it is a coinvariant with
respect to α.
P
Proof. Suppose that h is a coinvariant; that is,
h2 ⊗h3 S(h1P
) = h⊗1. By applyingP
∆⊗id
to this equality, and then applying the twist τ(132) , we obtain h3 ⊗h4 S(h1 )⊗h2 = h2 ⊗
2
P
P
1 ⊗ h1 . Next, by applying id ⊗ µ to this equality we obtain
h3 ⊗ h4 S(h1 )h2 = h2 ⊗ h1 .
Note that the right hand side of this last equality is τ ∆(h), whereas the left hand side is
X
X
X
X
h3 ⊗ h4 S(h1 )h2 =
h2 ⊗ h3 ε(h1 ) =
ε(h1 )h2 ⊗ h3 =
h1 ⊗ h2 = ∆(h),
as required.
P
P
Conversely, assume that ∆(h) = τ ∆(h); that
h1 ⊗ h2 =
hP
2 ⊗ h1 . By applying
P is,
∆ ⊗ id followed by S ⊗ id ⊗ id we obtain
S(h1 ) ⊗ h2 ⊗ h3 =
S(h2 ) ⊗ h3 ⊗ h1 .
Next, by applying the
P
P twist τ(132) to this equation, followed by applying id ⊗ µ, we obtain
h2 ⊗ h3 S(h1 ) = h3 ⊗ h1 S(h2 ).
The left hand side of the last equality is α(h), whereas the right hand side is
X
X
X
h3 ⊗ h1 S(h2 ) =
h2 ⊗ ε(h1 ) =
ε(h1 )h2 ⊗ 1 = h ⊗ 1;
and so h is a coinvariant.
Therefore Aα-co-A , the set of α-coinvariants is a subalgebra of A. However, A is not a
comodule algebra with respect to α. We have the following product formula.
Lemma 2.2 For any a, b ∈ A we have
X
α(ab) =
(a2 ⊗ a3 ) · α(b) · (1 ⊗ S(a1 )).
In particular, if b is an α-coinvariant, then α(ab) = α(a)α(b).
Proof. Using the fact that ∆(2) : A → A ⊗ A ⊗ A is an algebra homomorphism and that
S is an algebra anti-isomorphism we get
X
α(ab) =
(ab)2 ⊗ (ab)3 S((ab)1 )
XX
=
a2 b2 ⊗ a3 b3 S(b1 )S(a1 )
X
X
=
(a2 ⊗ a3 )(
b2 ⊗ b3 S(b1 ))(1 ⊗ S(a1 ))
which gives the desired formula. Moreover, if α(b) = b ⊗ 1, then α(b) and 1 ⊗ S(a 1 )
commute, and we get the second statement.
3
The conjugation coaction on quantum matrices
We define a quantum analogue of the conjugation action of the general linear group on the
space of N × N matrices.
It is easy to check (see [2]) that O(Mq (N, C)) is a left (respectively, right) comodule
algebra with respect to the left (respectively, right) comultiplications λ (respectively, ρ)
given by the formulae
λ : O(Mq (N, C)) → O(GLq (N, C)) ⊗ O(Mq (N, C)), xij 7→
3
N
X
s=1
uis ⊗ xsj
ρ : O(Mq (N, C)) → O(Mq (N, C)) ⊗ O(GLq (N, C)), xij 7→
N
X
s=1
xis ⊗ usj .
These two comultiplications commute in the sense that we have the equality (id ⊗ ρ) ◦ λ =
(λ ⊗ id) ◦ ρ. This equality follows from the coassociativity of the comultiplication ∆ on
O(GLq (N, C)), since the previous maps are identical to the restriction of (id ⊗ ∆) ◦ ∆ =
∆(2) = (∆ ⊗ id) ◦ ∆ to O(Mq (N, C)).
Now set
αq : O(Mq (N, C)) → O(Mq (N, C)) ⊗ O(GLq (N, C)),
αq := (id ⊗ µ) ◦ (ρ ⊗ id) ◦ τ ◦ (S ⊗ id) ◦ λ.
(2)
(Here S is the antipode and µ is the multiplication in O(GL q (N, C)), and τ is the flip.)
In the special case q = 1, it is obvious that α1 is the comorphism induced by
M (N, C) × GL(N, C) → M(N, C), (A, g) 7→ g −1 Ag.
The coaction αq defined in (2) coincides with the restriction to O(Mq (N, C)) of the
adjoint coaction (1) of the Hopf algebra O(GLq (N, C)) on itself; that is, using Sweedler’s
notation
X
αq (u) 7→
u2 ⊗ u3S(u1 ), u ∈ O(GLq (N, C)).
On the generators of O(Mq (N, C)) the coaction αq can be given explicitly as
αq (xij ) =
N
X
m,s=1
xms ⊗ usj S(uim ).
In the sequel αq will denote both the coactions of O(GLq (N, C)) on itself and on
O(Mq (N, C)), and we will omit the index q and write α for αq , provided that no confusion
is likely to arise.
As we noted in Section 2, O(Mq (N, C)) is not a comodule algebra with respect to α.
However, as an immediate corollary of Theorem 2.1 or Lemma 2.2 we get the following.
Proposition 3.1 The set of coinvariants O(M q (N, C))α-co-O(GLq (N,C)) is a subalgebra of
O(Mq (N, C)).
4
Construction of coinvariants
Fix an integer t with 1 ≤ t ≤ N. Let I and J be subsets of {1, . . . , N} with |I| = |J| = t.
The subalgebra of O(Mq (N, C)) generated by xij with i ∈ I and j ∈ J can be regarded as
an algebra of t × t quantum matrices, and so we can calculate its quantum determinant 4
this is a t × t quantum minor and we denote it by [I|J]. Recall that comultiplication of
quantum minors is given by the rule
X
∆([I|J]) =
[I|K] ⊗ [K|J],
|K|=t
see [6, (1.9)], so that
X
α([I|J]) =
|K|=|M |=t
[M|K] ⊗ [K|J]S([I|M ]).
The quantum minor [I|I] is said to be a principal quantum minor. We denote the sum of
all the principal quantum minors of a given size i by σ i . Note that σ1 = x11 + · · · + xN N
and that σN = detq . We denote the subalgebra of O(Mq (N, C)) generated by all of the σi
by Chσ1 , . . . , σN i.
Proposition 4.1
Chσ1 , . . . , σN i ⊆ O(Mq (N, C))α-co-O(GLq (N,C)) .
Proof. By Corollary 3.1, it is enough to show that each σ i is a coinvariant.
X
α(σi ) = α(
[I|I])
|I|=i
=

X
|I|=i
=

|K|=|M |=i
X
|K|=|M |=i
=
X
|K|=|M |=i
=
X
|K|=i
X
[M|K] ⊗ [K|I]S([I|M ])
[M|K] ⊗

X

[K|I]S([I|M])
|I|=i
[M |K] ⊗ ε([K|M ])






[K|K] ⊗ 1
= σi ⊗ 1
where the antepenultimate equality follows from µ ◦ (id ⊗ S) ◦ ∆ = ε, cf. [5, 1.5.2].
(Alternatively, one can check easily that σi is co-commutative, and refer to Theorem 2.1.)
Our aim is to show that, in the case that q is not a root of unity, Chσ 1 , . . . , σN i =
O(Mq (N, C))α-co-O(GLq (N,C)) and that this algebra is a commutative polynomial algebra
in the σi , i = 1 . . . , N. In order to do this, we need to recall the character theory of
corepresentations of cosemisimple Hopf algebras. This is done in the next section.
5
5
Corepresentation theory
We recall some basic facts about the corepresentation theory of the quantum general linear
group. These can be found, for example, in the book [3, Part III., 11.5.].
By a corepresentation of a Hopf algebra A on the vector space V we mean a right
A-comodule T : V → V ⊗ A. If V is finite dimensional, then we may define the character
χT as the element of A obtained by summing the diagonal elements of the matrix corepresentation belonging to T with respect to an
arbitrary basis of V . Given a Hopf algebra
homomorphism π : A → B we have that T B := (idV ⊗ π) ◦ T is a corepresentation of B
on V . If V is finite dimensional, then obviously we have χ = π(χT ).
T
B
∗
If q ∈ C is not a root of unity (or if q = 1), then O(GLq (N, C)) is a cosemisimple Hopf
algebra; that is, any corepresentation decomposes into a direct sum of irreducible corepresentations. The irreducible corepresentations are indexed by P + = {λ = (λ1 , . . . , λN ) ∈
ZN | λ1 ≥ · · · ≥ λN }, the set of integral dominant weights of the Lie algebra gl(N, C).
For λ ∈ P+ denote by Tqλ the associated finite dimensional irreducible corepresentation
of O(GLq (N, C)); for the definition of Tqλ see [3, page 439]. In the case that q is not a
root of unity or q = 1 the Tqλ (λ ∈ P+ ) provide a complete list of pairwise non-isomorphic
irreducible corepresentations of O(GLq (N, C)).
Let D be the diagonal subgroup of GL(N, C). Its coordinate Hopf algebra O(D) is the
−1
commutative algebra C[t1 , t−1
1 , . . . , tN , tN ] of Laurent polynomials with comultiplication
∆(ti ) = ti ⊗ ti and counit ε(ti ) = 1. There exists a surjective Hopf algebra homomorphism
πD : O(GLq (N, C)) → O(D) determined by πD (uij ) = δij ti , i, j = 1, . . . , N. Therefore we
say that D is a quantum
subgroup of GLq (N, C), called the diagonal subgroup.
λ
The character of Tq D does not depend on q. Moreover,
πD (χTqλ ) = Sλ (t1 , . . . , tN ) ∈ Z[t1 , . . . , tN ],
where Sλ (t1 , . . . , tN ) · (t1 · · · tN )−λN is the Schur polynomial belonging to the partition
(λ1 − λN , λ2 − λN , . . . , λN −1 −L
λN , 0) (see [4]).
Note that O(Mq (N, C)) = d∈N0 O(Mq (N, C))d is an algebra graded by the semigroup
of non-negativeL
integers, and the corepresentation α q is homogeneous; that is, α decomposes as αq = d∈N0 αqd , where αqd : O(Mq (N, C))d → O(Mq (N, C))d ⊗ O(GLq (N, C)) is
the restriction of αq to the degree d component.
Lemma 5.1 For any q ∈ C∗ and d ∈ N0 the O(D)-corepresentation αqd D is equivalent to
the O(D)-corepresentation α1d D .
Proof. Since
id ⊗ πD : O(GLq (N, C)) ⊗ O(GLq (N, C)) → O(GLq (N, C)) ⊗ O(D)
is an algebra homomorphism, it follows from Lemma 2.2 that
X
αq D (ab) =
(a2 ⊗ πD (a3 )) · αq D (b) · (1 ⊗ πD (S(a1 ))).
Now 1 ⊗ πD (S(a1 )) is a central element in O(GLq (N, C)) ⊗ O(D), so
X
αq D (ab) = (
a2 ⊗ πD (a3 )πD (S(a1 ))) · αq D (b) = αq D (a) · αq D (b).
6
So O(Mq (N, C)) becomes a right O(D)-comodule algebra under the coaction α q D (though
it was not necessarily an O(GLq (N, C))-comodule algebra under αq ). For an index set
−1
I = (i1 , j1 , . . . , id , jd ) ∈ {1, . . . , N } 2d set xI := xi1 j1 · · · xid jd and tI := t−1
i1 tj1 · · · tid tjd . By
I
I
I
the definition of α we have that α D (xij ) = xij ⊗ t−1
i tj , hence α D (x ) = x ⊗ t . For each
d there exists a subset I of {1, . . . , N}2d , independent of q, such that {xI | I ∈ I} is a
basis of O(Mq (N, C))d . Hence, O(Mq (N, C))d is the direct sum of the one dimensional
O(D)-sub-comodules CxI (I ∈ I), where the corepresentation of O(D) on CxI is given by
the character tI .
Denote by md (λ) the multiplicity of T1λ in the O(GL(N, C))-corepresentation O(M(N, C))d
under α1 , that is,
M
α1d ∼
md (λ)T1λ ,
=
λ∈P+
where mT := T ⊕· · ·⊕T (m direct summands) for a corepresentation T and a non-negative
integer m. Then passing to characters we have
X
md (λ)Sλ (t1 , . . . , tN ),
πD (χαd1 ) =
λ∈P+
and as an immediate corollary of Lemma 5.1 we get the following:
Corollary 5.2 For any q ∈ C∗ and d ∈ N0 we have
X
md (λ)Sλ (t1 , . . . , tN ).
πD (χαdq ) =
λ∈P+
If q is not a root of unity, then O(GLq (N, C)) is cosemisimple, and the decomposition
of the character of αqd into the sum of characters of irreducible corepresentations implies
the corresponding decomposition of the corepresentation itself, hence we get:
Corollary 5.3 If q ∈ C∗ is not a root of unity (or if q = 1) then for any d ∈ N 0 we have
M
αqd ∼
md (λ)Tqλ .
=
λ∈P+
Since the corepresentation αq is homogeneous, O(Mq (N, C))α-co-O(GLq (N,C)) is a graded
subalgebra of O(Mq (N, C)). Its Hilbert series H(O(Mq (N, C))α-co-O(GLq (N,C)) ; t) ∈ Z[[t]]
is the formal power series where the coefficient of td is the dimension of the degree d
component.
Proposition 5.4 If q ∈ C∗ is not a root of unity (or if q = 1), then
1
.
i)
(1
−
t
i=1
H(O(Mq (N, C))α-co-O(GLq (N,C)) ; t) = QN
7
Proof. Since O(GLq (N, C)) is cosemisimple, by the definitions of Hilbert series and coinvariants the dth coefficient of H(O(Mq (N, C))α-co-O(GLq (N,C)) ; t) is the multiplicity of the
(0,... ,0)
trivial corepresentation Tq
in O(Mq (N, C))d . By Corollary 5.3 this multiplicity does
not depend on q. In the case q = 1 the algebra O(M(N, C)) α-co-O(GL(N,C)) coincides with
the algebra of polynomial invariants of N × N matrices with respect to the conjugation
action
linear group. It is well known that the Hilbert series of this algebra
QN of the igeneral
−1
is i=1 (1 − t ) .
6
Main results
In this section we show that the coinvariants σi generate the algebra of coinvariants for the
conjugation coaction, and that this algebra of coinvariants is in fact a commutative polynomial algebra in the σi . First, some simplifying notation. We set A := O(Mq (N, C)), B :=
O(Mq (N, C))α-co-O(GLq (N,C)) and C := Chσ1 , . . . , σN i. Thus, C ⊆ B ⊆ A. The first thing to
notice is that these are inclusions of graded algebras. This is because the coinvariants σ i are
homogeneous elements of degree i in O(Mq (N, C)), and so Chσ1 , . . . , σN i inherits a graded
algebra structure from O(Mq (N, C)). The coaction α sends the subspace O(Mq (N, C))d
of elements of O(Mq (N, C)) that have degree d into O(Mq (N, C))d ⊗ O(GLq (N, C)),
and so O(Mq (N, C))α-co-O(GLq (N,C)) has a graded algebra structure with respect to which
Chσ1 , . . . , σN i is a graded subalgebra.
P i
P
We define a partial order on Z[[t]] in the following way. Let f =
fi t and g = gi ti
be elements of Z[[t]]. Then, set f ≤ g if and only if f i ≤ gi for each i.
If B is a graded subalgebra of a graded algebra A then H(B, t) ≤ H(A, t), with equality
if and only if B = A. Similarly, if I is a homogeneous ideal of A then H(A/I, t) ≤ H(A, t),
with equality if and only if I = 0.
Theorem 6.1 If q ∈ C∗ is not a root of unity (or if q = 1), then
O(Mq (N, C))α-co-O(GLq (N,C)) = Chσ1 , . . . , σN i
and this algebra of coinvariants is a commutative polynomial algebra of dimension N.
Proof.
The above inclusions immediately yield that
H(C, t) ≤ H(B, t) = QN
1
i
i=1 (1 − t )
,
where the equality comes from Proposition 5.4.
The elements xij , i 6= j, generate an ideal I of A = O(Mq (N, C)) such that the factor
algebra D := A/I is a commutative polynomial algebra in N indeterminates, the images
of x11 , . . . , xN N . Let us set Xi to be the image of xii and consider D to be graded with Xi
given degree 1. Thus, the natural map π of A onto D is a graded homomorphism. Let φ
be the restriction of this map to the graded subalgebra C. Note that φ(σ i ) = si , where si
is the ith elementary symmetric polynomial in the Xj . Then Im(φ) = C[s1 , . . . , sN ]. It is
Q
i −1
well-known that H(C[s1 , . . . , sN ], t) = N
i=1 (1 − t ) , so this epimorphism yields
N
Y
i=1
(1 − ti )−1 = H(C/ ker(φ), t) ≤ H(C, t).
8
Together with the earlier inequality, this gives
H(C/ ker(φ), t) = H(C, t) = H(B, t) =
N
Y
i=1
(1 − ti )−1
which forces C = B, as required, and also forces φ to be an isomorphism. In particular,
B=C∼
= C[s1 , . . . , sN ] is a commutative polynomial algebra.
As applications of this result, we have the following description of the co-commutative
elements of O(Mq (N, C)) and O(GLq (N, C)) which we have not been able to obtain directly.
Corollary 6.2 If q ∈ C∗ is not a root of unity (or if q = 1), then the subalgebra
of co-commutive elements of O(Mq (N, C)) is Chσ1 , . . . , σN i, and, consequently, the cocommutative elements form a commutative polynomial algebra of dimension N.
Proof. This follows immediately from the previous theorem and Theorem 2.1.
It is easy to derive from Theorem 6.1 a description of the α-coinvariants (or in other
words, the co-commutative elements) in the Hopf algebra O(GL q (N, C)):
Corollary 6.3 If q ∈ C∗ is not a root of unity (or if q = 1), then
−1
O(GLq (N, C))α-co-O(GLq (N,C)) = Chσ1 , . . . , σN −1 , σN , σN
i
and this algebra of coinvariants (and hence the subalgebra of co-commutative elements)
is a one-variable Laurent polynomial algebra over the commutative polynomial algebra of
dimension N − 1.
d
Proof. For any h ∈ O(GLq (N, C)) there is an exponent d such that hσN
is contained in
O(Mq (N, C)). Since σN is a coinvariant, we have by Lemma 2.2 that h is a coinvariant if
d
is a coinvariant. The claim follows from this observation and Theorem 6.1.
and only if hσN
Remark 6.4 We expect that the same results hold also when q is a root of unity. In fact,
we can derive the conclusion that Chσ1 , . . . , σN i is a commutative polynomial subalgebra
of the algebra of coinvariants in the general case from our restricted result. However, we
cannot at the moment show that these are all of the coinvariants. The cosemisimplicity of
O(GLq (N, C)) for q generic was used in Corollary 5.3. We would need another argument
to derive Proposition 5.4 from Corollary 5.2, when q is a root of unity. Such kind of
replacement of the semisimplicity is provided by the theory of tilting modules (or modules
with good filtration), when invariants of the classical conjugation action are studied over
a positive characteristic base field.
Remark 6.5 In [9], Zhang proves two versions of a Quantum Cayley-Hamilton Theorem.
The coefficients that arise in the identity [9, Theorem 2.4] are the σ i that we have seen are
the basic coinvariants for the conjugation coaction α.
Remark 6.6 We plan to consider the problem of determining the coinvariants of several
quantum matrices under the conjugation coaction in a later paper.
9
7
Other versions of the adjoint coaction
There is an element of choice in the definition (1) of the adjoint coaction. In fact, there
are four ‘obvious’ choices for the adjoint coaction of a Hopf algebra A on itself, two right
coactions (α,β) and two left coactions (γ, δ). These are:
P
(i) α(h) := h2 ⊗ h3 S(h1 )
P
(ii) β(h) := h2 ⊗ S(h1 )h3
P
(iii) γ(h) :=
h1 S(h3 ) ⊗ h2 , and
P
(iv) δ(h) := S(h3 )h1 ⊗ h2 .
Any of them can be used to define a coaction of O(GL q (N, C)) on O(Mq (N, C)), in
order to obtain a quantum analogue of the conjugation action of GL(N, C) on M (N, C).
We have chosen case (i). The case of the corresponding left coaction (iv) yields no essential
difference, because similarly to Theorem 2.1 we have that h is a δ-coinvariant if and only
if h is co-commutative, so Theorem 6.1 remains valid if we replace α by δ.
On the other hand, in all the literature we checked the adjoint coaction is defined to be
β
or
only if ∆(h) =
P 2γ. It is shown in [1, Lemma 1.1] that h ∈ A is a γ-coinvariant if and P
S (h2 ) ⊗ h1 . Similarly, h ∈ A is a β-coinvariant if and only if ∆(h) =
h2 ⊗ S 2 (h1 ).
Since the square of the antipode of O(GLq (N, C)) is not the identity if q 2 6= 1, the sets of
coinvariants in O(Mq (N, C)) with respect to β or γ are not the same as for α.
For i = 1, . . . , N consider the weighted trace functions
X
τi :=
q −2|I| [I|I],
I
ρi :=
X
q 2|I| [I|I].
I
Here, and elsewhere, all summations are over index sets of a fixed size (eg. i in the case of
τi ). In exponents, we will use |I| to denote the sum of the entries of the index set I.
First, we need to calculate the effect of S 2 on quantum minors.
Lemma 7.1 The action of S 2 on quantum minors is given by
S 2 ([I|J]) = q 2(|I|−|J|) [I|J]
and, consequently,
[I|J] = S 2 (q 2(|J|−|I|) [I|J]).
N
−1
Proof.This follows from the formula S 2 ((uij )N
i,j=1 ) = Q · (uij )i,j=1 · Q , where Q is the
diagonal matrix with diagonal entries 1, q 2 , . . . , q 2(N −1) (see for example [8, Theorem 4]).
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Proposition 7.2 The weighted sum of principal quantum minors τ i is a coinvariant for
the coaction β, and ρi is a coinvariant for γ.
Proof. Note that
β([I|J]) =
X
K,M
[K|M] ⊗ S([I|K])[M|J],
hence
β(τi ) =
X
I
=
X
K,M
=
X
K,M
=
X
K,M
=
X
K,M
=
X
M
(
X
K,M
q −2|I| [K|M ] ⊗ S([I|K])[M|I]
[K|M] ⊗
[K|M] ⊗
[K|M] ⊗
(
X
q −2|I| S([I|K])[M|I]
I
(
X
)
)
q −2|I| S([I|K])S 2 (q 2(|I|−|M |) [M|I])
I
(
q
−2|M |
S
X
S([M|I])[I|K]
I
!)
)
q −2|M | [K|M] ⊗ S(ε([M |K]))
q −2|M | [M|M] ⊗ 1
= τi ⊗ 1.
Similar calculation shows that ρi is a coinvariant for γ.
Finally, we give the counterpart of Theorem 6.1 for the coactions β and γ.
Theorem 7.3 If q ∈ C∗ is not a root of unity, then
O(Mq (N, C))β-co-O(GLq (N,C)) = Chτ1 , . . . , τN i,
O(Mq (N, C))γ -co-O(GLq (N,C)) = Chρ1 , . . . , ρN i,
and both algebras of coinvariants are commutative polynomial algebras of dimension N.
Proof. We deal with β, the case of γ is the same. Similarly to the proof of Lemma 5.1
it is possible to show that we have the isomorphism of corepresentations β qd D ∼
= αqd D .
Therefore the algebra of β-coinvariants has the same Hilbert series as the commutative
polynomial algebra with generators of degree 1, 2, . . . , N. Using this one concludes the
result from Proposition 7.2 in the same way as in the proof of Theorem 6.1.
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References
[1] M. Cohen and S. Zhu, Invariants of the adjoint coaction and Yetter-Drinfeld categories,
J. Pure Appl. Alg. 159 (2001), 149–171.
[2] K. R. Goodearl, T. H. Lenagan and L. Rigal, The first fundamental theorem of coinvariant theory for the quantum general linear group, Publ. Res. Inst. Math. Sci., Kyoto
Univ. 36, No. 2 (2000), 269–296.
[3] A. Klimyk and K. Schmüdgen, “Quantum Groups and Their Representations”,
Springer-Verlag, Berlin, Heidelberg, New York, 1997.
[4] I. G. Macdonald, “Symmetric Functions and Hall Polynomials”, 2nd Ed., Clarendon
Press, Oxford, 1995.
[5] S. Montgomery, “Hopf algebras and their actions on rings”, American Mathematical
Society, Providence, 1993.
[6] M. Noumi, H. Yamada, and K. Mimachi, Finite dimensional representations of the
quantum group GLq (n; C) and the zonal spherical functions on Uq (n − 1)\Uq (n),
Japanese J. Math. 19 (1993), 31-80.
[7] B. Parshall and J. Wang. Quantum linear groups. Mem. Amer. Math. Soc 89 (1991),
no.439.
[8] N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups
and Lie algebras, (Russian) Algebra i Analiz 1 (1989), 178–206.
[9] J. J. Zhang, The quantum Cayley-Hamilton Theorem, J. Pure and App. Alg. 129,
(1998), 101-109.
Rényi Institute of Mathematics, Hungarian Academy of
Sciences, P.O. Box 127, 1364 Budapest, Hungary
E-mail: [email protected], [email protected]
Department of Mathematics, University of Edinburgh,
James Clerk Maxwell Building, King’s Buildings,
Mayfield Road, Edinburgh EH9 3JZ, Scotland
E-mail: [email protected]
(Domokos is in Edinburgh until February 2004.)
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